Theorem frgrncvvdeqlem1 28663

Description: Lemma 1 for frgrncvvdeq 28673. (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 8-May-2021.) (Proof shortened by AV, 12-Feb-2022.)

Hypotheses

Ref	Expression
frgrncvvdeq.v1	⊢ 𝑉 = (Vtx‘𝐺)
frgrncvvdeq.e	⊢ 𝐸 = (Edg‘𝐺)
frgrncvvdeq.nx	⊢ 𝐷 = (𝐺 NeighbVtx 𝑋)
frgrncvvdeq.ny	⊢ 𝑁 = (𝐺 NeighbVtx 𝑌)
frgrncvvdeq.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
frgrncvvdeq.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
frgrncvvdeq.ne	⊢ (𝜑 → 𝑋 ≠ 𝑌)
frgrncvvdeq.xy	⊢ (𝜑 → 𝑌 ∉ 𝐷)
frgrncvvdeq.f	⊢ (𝜑 → 𝐺 ∈ FriendGraph )
frgrncvvdeq.a	⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸))

Assertion

Ref	Expression
frgrncvvdeqlem1	⊢ (𝜑 → 𝑋 ∉ 𝑁)

Proof of Theorem frgrncvvdeqlem1

Step	Hyp	Ref	Expression
1		frgrncvvdeq.xy	. . . 4 ⊢ (𝜑 → 𝑌 ∉ 𝐷)
2		df-nel 3050	. . . . 5 ⊢ (𝑌 ∉ 𝐷 ↔ ¬ 𝑌 ∈ 𝐷)
3		frgrncvvdeq.nx	. . . . . 6 ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋)
4	3	eleq2i 2830	. . . . 5 ⊢ (𝑌 ∈ 𝐷 ↔ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
5	2, 4	xchbinx 334	. . . 4 ⊢ (𝑌 ∉ 𝐷 ↔ ¬ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
6	1, 5	sylib 217	. . 3 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
7		nbgrsym 27730	. . 3 ⊢ (𝑋 ∈ (𝐺 NeighbVtx 𝑌) ↔ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
8	6, 7	sylnibr 329	. 2 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
9		frgrncvvdeq.ny	. . . 4 ⊢ 𝑁 = (𝐺 NeighbVtx 𝑌)
10		neleq2 3055	. . . 4 ⊢ (𝑁 = (𝐺 NeighbVtx 𝑌) → (𝑋 ∉ 𝑁 ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑌)))
11	9, 10	ax-mp 5	. . 3 ⊢ (𝑋 ∉ 𝑁 ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑌))
12		df-nel 3050	. . 3 ⊢ (𝑋 ∉ (𝐺 NeighbVtx 𝑌) ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
13	11, 12	bitri 274	. 2 ⊢ (𝑋 ∉ 𝑁 ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
14	8, 13	sylibr 233	1 ⊢ (𝜑 → 𝑋 ∉ 𝑁)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2106   ≠ wne 2943   ∉ wnel 3049  {cpr 4563   ↦ cmpt 5157  ‘cfv 6433  ℩crio 7231  (class class class)co 7275  Vtxcvtx 27366  Edgcedg 27417   NeighbVtx cnbgr 27699   FriendGraph cfrgr 28622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-nbgr 27700

This theorem is referenced by:  frgrncvvdeqlem7  28669  frgrncvvdeqlem8  28670  frgrncvvdeqlem9  28671